$\begingroup$Generally speaking, the problem of determining the optimal tradeoff between the parameters $T,R,N$ is very hard, so you shouldn't expect any simple algorithm that works in general.$\endgroup$
– Yuval FilmusJun 3 at 12:36

$\begingroup$Thank you. I've come across several proofs, where they use a probabilistic approach. But I still don't understand how to build the code from scratch. There are some methods, as they call them "Building codes from other codes", such as the parity check bit method or the puncturing method, which are based upon some already existing codes. The methods are discussed in the Algorithmic Introduction to Coding Theory lectures by M. Sudan.$\endgroup$
– Evgeny MamaevJun 3 at 13:57

$\begingroup$Coding theory is a vast topic. Entire books have been written on it. This makes your question quite broad.$\endgroup$
– Yuval FilmusJun 3 at 13:59

$\begingroup$The task I'm trying to accomplish is to write an algorithm, which accepts three numbers, named T, R, N, and shows the generator matrix of the code as a result. Let it be an arbitrary code without any specific constraints on the input parameters.$\endgroup$
– Evgeny MamaevJun 3 at 14:09

1 Answer
1

Look at Reed–Solomon codes. They are MDS codes, and AFAIK, MDS codes are most efficient codes that operates on independent blocks of bits.

If you need to support only erasures, there is simple to describe algorithm - consider input words as values of some polynomial at points 0..n-1, and compute (using a Galois Field) values of the same polynomial at points n..n+m-1 - it will be possible to restore polynomial coefficients from any n survived values.

$\begingroup$Thank you for the article of the erasure-only codes, you should definitely translate it to English. I liked its conciseness in Russian. However, I have not accepted your answer, since I had not found the answer in the article on how to construct the code.$\endgroup$
– Evgeny MamaevJun 4 at 6:32